Waves excited at a free surface in a half-infinite isotropic nonlinear elasic medium

Abstract

In nonlinear elastic equations, there exist two kinds of simple waves, i.e., Non-Coupled and Coupled Simple Waves which are alias named nonlinear P and S waves, respectively. Nonlinear equation used in this paper is of second order with respect to displacement and, as the result, used stress is also of second order. A used model is a two-dimensional one. The used stress condition is truely nonlinear instead of quasi-linear, where the latter is usually solved by use of perturbation procedure. In the model of a half-infinite elastic medium, the nonlinear stress-free surface condition is, as the first approximation, separated into three branches of stress conditions in the case of a direct-hit earthquake. For the incidence of Simple (nonlinear) Waves on the free surface, the reflected waves are evaluated for the above three branches of condition. The energy expression is then obtained in a weakly nonlinear case (the elastic constants associated with higher order terms are negligibly small). Among three branches of condition, the first one is a condition which causes ordinary reflected waves similar to P and S waves in a linear theory. The other two are conditions which produce retarded reflected waves with slow velocity near the free surface. As the result of the retardation, high energy flux occurs along the free surface. This high energy causes a large disaster on the occasion of large earthquake. When nonlinear P wave is retarded near the free surface, Rayleigh-type nonlinear wave appears along the free surface. The wave is then propagated at a velocity a little less than that of linear S wave. This behavior indicates the generation process of Rayleigh wave. These results are obtained by use of truely nonlinear stress condition at the free surface instead of quasi-linear one based on perturbation procedure.